Recent studies have shown that deductive reasoning skills are related to mathematical abilities. Nevertheless, so far the links between mathematical abilities and these two forms of deductive inference have not been investigated in a single study. It is also unclear whether these inference forms are related to both basic maths skills and mathematical reasoning, and whether these relationships still hold if the effects of fluid intelligence are controlled. We conducted a study with 87 adult participants. The results showed that transitive (...) reasoning skills were related to performance on a number line task, and conditional inferences were related to arithmetic skills. Additionally, both types of deductive inference were related to mathematical reasoning skills, although transitive and conditional reasoning ability were unrelated. Our results also highlighted the important role that ordering abilities play in mathematical reasoning, extending findings regarding the role of ordering abilities in basic maths skills. These results have implications for the theories of mathematical and deductive reasoning, and they could inspire the development of novel educational interventions. (shrink)

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

Math anxiety results in a drop in performance on various math-related tasks, including the symbolic number ordering task in which participants decide whether a triplet of digits is presented in order (e.g. 3-5-7) or not (e.g. 3-7-5). We investigated whether the strategy repertoire and reaction times during a symbolic ordering task were affected by math anxiety. In study 1, participants performed an untimed symbolic number ordering task and indicated the strategy they used on a trial-by-trial basis. The use of the (...) memory retrieval strategy, based on the immediate recognition of the triplet, decreased with high math anxiety, but disappeared when controlling for general anxiety. In the study 2, participants completed a timed version of the number order task. High math-anxious participants used the decomposition strategy (e.g. 5 is larger than 3 and 7 is larger than 5 to decide whether 3-5-7 is in the correct order) more often, and were slower in responding when both memory- and other decomposition strategies were used. Altogether, both studies demonstrate that high-math anxious participants are not only slower to decide whether a number triplet is in the correct order, but also rely more on procedural strategies. (shrink)

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A third hypothesis, (...) in line with dual-systems theory (e.g., Kahneman and Frederick, 2002 ) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others. (shrink)

This study aimed to investigate whether the ordinal judgments of high math-anxious and low math-anxious individuals differ. Two groups of 20 participants with extreme scores on the Shortened Mathematics Anxiety Rating Scale had to decide whether a triplet of numbers was presented in ascending order. Triplets could contain one-digit or two-digit numbers and be formed by consecutive numbers, numbers with a constant distance of two or three or numbers with variable distances between them. All these triplets were also presented unordered: (...) sequence order in these trials could be broken at the second or third number. A reverse distance effect of equal size was found in both anxiety groups. However, HMA participants made more judgment errors than their LMA peers when they judged one-digit counting ordered triplets. This effect was related to worse performance of HMA individuals on a symmetry span test and might be related to group differences on working memory. Importantly, HMAs were less accurate than LMA participants at rejecting unordered D2 sequences. This result is interpreted in terms of worse cognitive flexibility in HMA individuals. (shrink)

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children’s math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named ‘123 Bakery’ which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic and symbol-to-numerosity mapping. Fifty-six first graders (...) were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post- training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested. (shrink)

The proposal by Clarke and Beck offers a new explanation for the association between the approximate number system and math. Previous explanations have largely relied on developmental arguments, an underspecified notion of the ANS as an “error detection mechanism,” or affective factors. The proposal that the ANS represents rational numbers suggests that it may directly support a broader range of math skills.

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial—accessing underlying magnitude representation of symbols (i.e., symbol‐magnitude associations), processing relative order of symbols (i.e., symbol‐symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots‐number word matching task—thought to be a measure of symbol‐magnitude associations (numerical magnitude processing)—a numeral‐ordering task that focuses on (...) symbol‐symbol associations (numerical order processing), and a digit‐number word matching task targeting symbolic processing per se. Results showed that both numerical magnitude and order processing were uniquely related to arithmetic achievement, beyond the effects of domain‐general factors (intellectual ability, working memory, inhibitory control, and non‐numerical ordering). Importantly, results were different when a general measure of mathematics achievement was considered. Those mechanisms of symbolic number processing did not contribute to math achievement. Furthermore, a path analysis revealed that numerical magnitude and order processing might draw on a common mechanism. Each process explained a portion of the relation of the other with arithmetic (but not with a general measure of math achievement). These findings are consistent with the notion that adults’ arithmetic skills build upon symbol‐magnitude associations, and they highlight the effects that different math measures have in the study of numerical cognition. (shrink)